One can not help but note that Avraham and Sarah both laughed, yet El-him tells Avraham, “… and you shall call his name Yitzchaq….” (B’rashith [Genesis] 17.19).
“But My covenant will I establish with Isaac ….” (B’rashith [Genesis] 17.21).
“Cast out this bondwoman and her son, for the son of this bondwoman shall not inherit with my son, with Yitzchaq.” (B’rashith [Genesis] 21.10).
Yitzchaq means derision (laughter) or scorn as in mockery (Shoftim [Judges] 16.25, Tehillim [Psalms] 1.1, 2.4).
Avraham laughed outwardly (with the mouth, not internally) with joy of heart and in astonishment or wonder believed; while Sarah “snickered” internally!
Rabbi Naftali Zvi Yehuda Berlin of Volozhin, Ha’amek Davar: Commentary on the Pentateuch, New York: Friedman. <https://www.lookstein.org/professional-dev/and-sarah-laughed/>
The significance of this particular detail (Sarah’s denial; she was afraid, because her laughter and eaves-dropping were revealed) and what Moshe (Moses) is trying to convey here is that nothing is hidden from HaShem (18.14) and HaShem reveals to His Prophets what He is doing (18.17; Devarim [Deuteronomy] 29.29: “the secret things belong to HaShem […] but the revealed things belong to us [….];” Amos 3.7); in this case HaShem excluded from His Covenant (e.g. Yishmael) those who mock “the sign of His Covenant” which is circumcision. Genesis Rabbah 55.4
The covenant is not established with mockery (scorn), but with laughter of joy as Sarah said, “[…] all who hear will laugh with me.” 21.6 Sarah was a Prophetess as it says, “because of the words of Sarah.” 20.18
In fact, King David prophesied in Tehillim 105.9 that Yitzchaq’s name would be changed to Yischaq (laughter of the mouth).
The difference between Avraham’s laughter and Sarah’s is that Avraham “fell on his face” (he worshiped in belief) while Sarah “laughed within herself” in denial.
Where it says in 21.10, “with my son, with Yitzchaq (laughter)” the phrase, Im Yitzchaq is superfluous; meaning we are not to read her prophetic decree or statement as a proper name (with Isaac) but as a statement of law! The inheritance comes from proper treatment of our fellow-being and not with mockery or scorn as Yishmael was seen doing.